Generalization of Gamma Function and Its Applications to q-analysis. Michitomo Nishizawa

Transcription

1 Generalization of Gamma Function and Its Applications to q-analysis Michitomo Nishizawa

2 Preface The gamma function was introduced in the era of Euler and Gauss. This function had played fundamental roles in the theory of special functions. Basic properties of the gamma function are well known in the theory of classical analysis. For example, we can see that it has infinite product representations, that it satisfies the Bohr-Morellup theorem and that it closely related the Hurwitz zeta-function. It would be natural to long for giving reasonable generalization of the gamma function which satisfy the above properties. We remark two kind of generalization, with which we concern in this thesis. One is multiplization, the other is q-analogue. Multiple gamma function was introduced by E.W.Barnes [Bar99], [Bar00], [Bar0], [Bar04] in the end of 9 th century. He generalized the representation by the Hurwitz zeta-function. Hardy [Har05], [Har06] studied this function in relevant with his studies on elliptic functions. Shintani [Shi77], [Shi80] used it in his papers about the Kronecker limit formulas for a certain algebraic field. In 70 s, M.F.Vignéras [Vig79] redefined her multiple gamma function through generalization of the Bohr-Morellup theorem. This a special case of Barnes multiple gamma function. It appears in determinant representations of Laplacian in spectral geometry [Var88], [Vor87]. On the other hand, q-analogue of the gamma function the q-gamma function was introduced by Jackson [Jac04], [Jac05]. As same as the gamma function, the q-gamma function is significant in studies on q-analysis and important results are derived. R.Askey [Ask78] pointed out that the q-gamma function satisfies a q-analogue of the Bohr-Morellup theorem. Moak [Moa84] derived a q-analogue of the Stirling formula. These properties are generalize in this thesis. From the 80 s, geometrical interpretation of q-analysis has been recognized through studies on quantum groups. It also suggests a relation between integrable systems and q-analysis. From this point, it is expected that generalization of gamma functions should be important in future. This thesis consists of three parts. In Part I, we study a q-analogue of the Vignéras multiple gamma function the multiple q-gamma function and its classical limit. The relation between q-analogues of the multiple gamma functions and quantum integral systems are suggested by Freud and Zabrodin [FZ92]. However, they didn t give explicit form of this function in their paper. Motivated by Vignéras result [Vig79] and by Askey s result [Ask78], we construct a hierarchy of functions which represented by Weierstrass infinite product forms. We prove that they satisfy an q-analogue of the generalized Bohr-Morellup theorem and that these functions have infinite product representations corresponding to the Euler and the Gauss infinite product representation of the gamma function. Next, we investigate its classical limit. By means of the Euleri

3 MacLaurin summation formula, we can see that the logarithm of the multiple q-gamma function is represented by finite sum and that each terms of the sum converge uniformly as q. Namely, the classical limit satisfies the generalized Bohr-Morellup theorem and coincide with the Vignéras multiple gamma function. Furthermore, this result informs us some interesting properties of the Vignéras multiple functions. By transforming the representation of this limit, we derive an asymptotic behavior a generalization of the Stirling formula and an infinite product representation a generalization of the Weierstrass product representation of the Vignéras multiple gamma function. In Part II, we construct a q-analogue of the Hurwitz zeta-function q-hurwitz zeta-function. The Hurwitz zeta-function is closely related the gamma function. It is expected that the q-hurwitz zeta-function is related to the q-gamma function, too. We start from an observation on spectra of the Casimir operator on the quantum group SU q 2. Similarly to the theory of spectral zeta-functions, we give a definition of a q-analogue of the Hurwitz zeta-function. It is different from Cherednik s q-zeta-function [Che88]. First, we investigate several properties of the q-hurwitz zeta-function as an meromorphic function. For example, location of poles, Laurent expansion at s 0 and so on. Next, we apply the Euler-MacLaurin summation formula to the q-hurwitz zeta-function. As a result, a representation of the q-hurwitz zeta-function, containing the Gauss hypergeometric function, is obtained. We remark that its classical limit represents new representation formula of the classical Hurwitz zeta-function. Applying the Euler-MacLaurin summation formula, we also obtain an analogue of the Hurwitz relation. The Hurwitz relation represents the functional relation of the Riemann zeta-function when z. It is expected that its q-analogue plays fundamental role in the q-zeta function theory. Furthermore, we consider relation between the q-hurwitz zeta-function and the q-gamma function. As same as the classical theory, we can obtain a representation formula of the q-gamma function by using the q-hurwitz zeta-function. Finally, some related topics are discussed. In Part III, we apply the Barnes multiple gamma functions to q-analysis, especially in the case when q. In this case, we can not define q-gamma function in similar way to the case when 0 < q <. It is necessary to construct a solution satisfying the same functional relation as the q-gamma function with 0 < q <. For this end, we used Kurokawa s multiple sine function, which obtained from Barnes multiple gamma function by means of complementary formula. First, we construct two kind of solution of a q-difference analogue of Gauss hypergeometric equation with q. One is a solution of the Barnes type, the other is the Euler type. Each type solutions can be generalized. We construct a Barens type solution of a q-difference analogue of generalized hypergeometric equation r F s with q. On the other hand, the method of Euler type integration can be applied to a q-difference analogue of Lauricella s D-type hypergeometric equation. Next, we study contiguity relations of these integral solutions. In the case when 0 < q <, contiguous relations are studied on for various hypergeometric q-difference systems [DH97], [Hor92], [Nak95], [NN], [Nou92]. As same as the case when 0 < q <, these solutions satisfy contiguity relations. However, relation between these formulas and representation theory is left unclear because we have to impose conditions on parameters of the hypergeometric functions. ii

4 It is the authors pleasure to gratitude people who support him. First, he expresses his deep gratitude to his parents and his brother. He also thanks Sin-Ichi Azuma, Takashi Morimoto, Masaki Wakabayashi, Akio Tokuda, Yuji Hasegawa, Teruaki Kadokura, Hideaki Morita, Yoshihisa Saito, Satoshi Tsujimoto, Yasushi Homma and Hiroshi Yaoko for their heartfelt encouragements and for joyful discussions with them. Last but not least, he expresses his deep gratitude to Professor Kimio Ueno for his encouragements and for his valuable suggestions. Michitomo Nishizawa iii

5 Contents Preface i I Generalization of Gamma Function vi A survey of the multiple gamma function and the multiple q- gamma function. The gamma function The Barnes G-function The Barnes multiple gamma function The Vignéras multiple gamma function The q-gamma function A q-analogue of the multiple gamma function 6 2. Construction of a q-analogue of the multiple gamma functions Infinite product representation of G n z ; q The Classical Limit of the Multiple q-gamma Functions 0 3. The Euler-MacLaurin expansion of G n z ; q The classical limit of G n z ; q and the asymptotic expansion of G n z The classical limit of G n z ; q Asymptotic expansion of G n z The Weierstrass product representation for the G n z Rewriting the Euler-MacLaurin expansion of G n z An infinite product representation for the ζ j Good representation for K j z A proof of main theorem Examples of the Weierstrass product representation for G n z II A q-analogue of the Hurwitz Zeta-Function 39 4 A q-analogue of the Hurwitz Zeta-Function Observation from the theory of the quantum group SU q The q-hurwitz zeta-function The Euler-MacLaurin expansion Classical limit iv

6 4.5 A q-analogue of the functional equation for the Riemann zetafunction Stirling s formula for the q-gamma function and dilogarithm Miscellaneous topics Integral representation of q-hurwitz zeta-function Integral representation of q-riemann zeta-function at positive integer III Application to q-analysis with q 5 5 Introduction 52 6 Preliminaries Gauss hypergeometric function Barnes contour integral representation Euler s integral representation The hypergeometric q-difference equations Contiguity relations Generalized hypergeometric series r F r Lauricella s D-type hypergeometric function and its q-analogue Kurokawa s double sine function and a q-shifted factorial with q Kurokawa s double sine function q-shifted factorial with q Integral solutions of the Barnes type with q Solution of a q-difference analogue of Gauss hypergeometric equation Definition of the integral The hypergeometric q-difference equation with q Contiguity relation for Φz Solution of generalized hypergeometric equation with q Integral solutions of the Euler type with q A solution of q-difference analogue of Gauss hypergeometric equation Definition of the integral The difference equation for Ψx Solution of a q-difference analogue of Lauricella s D-type hypergeometric equation Construction of a Solution Contiguity Operators Acting on Ψ D x v

7 Part I Generalization of Gamma Function vi

8 Chapter A survey of the multiple gamma function and the multiple q-gamma function. The gamma function The following are well-known facts in the classical analysis: The Bohr-Morellup theorem says that the gamma function Γz is characterized by the three conditions, Γz zγz, 2 Γ, 3 d 2 log Γz 0 for z 0. dz2 The gamma function is meromorphic on C, and has an infinite product representation { Γz e γx z } z. e n, n.2.3 n N! Γz lim N z z 2 z N N z, { Γz z z }, n n n where γ is the Euler constant. These are usually called the Weierstrass product representation, the Gauss product representation and the Euler product representation, respectively. Another representation of the gamma function is derived from the Hurwitz zeta function :.4 ζs, z : z k s for Rs >. k0

9 It is well-known that Γz 2π expζ 0, z, where ζ 0, z d ds ζs, z s0. The gamma function has an asymptotic expansion formula, i.e. the Stirling formula, log Γz z logz z ζ 0 2 r B 2r [2r] 2 z 2r, as z in a sector δ : {z C arg z < π δ} 0 < δ < π, where ze tz e z n0 B n t z n, n! B k B k 0 the Bernoulli number, ζs is the Riemann zeta function, ζ s d ds ζs and [x] r xx x r. Note that ζ 0 log 2π..2 The Barnes G-function Barnes [Bar99] introduced the function Gz which satisfies Gz ΓzGz, 2 G, 3 d 3 log Gz 0 for z 0, dz3 and he called this the G-function. He proved that the G-function has an infinite product representation. Gz e zζ 0 z2 2 γ z2 z 2 and an asymptotic expansion.5 log Gz k { z } k exp z z2 k 2k z 2 2 logz z2 z 2 3 zζ 0 log A O z as z in the sector δ, where A is called the Kinkelin constant. Voros showed this constant can be written with the first derivative of the Riemann zeta function cf [Vor87], [Var88] log A ζ 2. 2

10 .3 The Barnes multiple gamma function We assume that ω, ω 2,, ω n lie on the same side of some straight line through the origin on the complex plane. The Barnes zeta function [Bar04] is defined as ζ n s, z; ω : k,k 2,,k n0 z k ω k n ω n s. where ω : ω, ω 2,, ω n. This is a generalization of the Hurwitz zeta-function. As a generalization of the formula.4, Barnes [Bar04] introduced his multiple gamma functions through Γ n z, ω : expζ ρ n ω n0, z; ω. where log ρ n ω : lim z 0 [ζ n0, z; ω log z]. It is easy to see that Γ n z, ω satisfies the functional relation Γ n z, ω Γ n z ω i, ω ρ n ωi Γ n z, ωi, where ωi : ω, ω i, ω i,, ω n..4 The Vignéras multiple gamma function As a generalization of the gamma function and the G-function, Vignéras [Vig79] introduced a hierarchy of functions which she called the multiple gamma functions Theorem.4. There exists a unique hierarchy of functions which satisfy G n z G n zg n z, 2 G n, 3 d n dz n log G nz 0 for z 0, 4 G 0 z z. Applying Dufresnoy and Pisot s results [DP63], she showed that these functions satisfying the above properties are uniquely determined and that G n z has an infinite product representation.6 G n z exp [ m N n N ze n n h p h z h! ] ψ h n 0 Eh n [ z n { n n l }] n l z exp, sm n l sm l0 3

11 where E n z : m N n N [{ n l0 ψ n z : log G n z, e tz e z : p k t zk k! k0 n l n l } n l z n log z ], sm sm sm : m m 2 m n for m m, m 2,, m n, and N N {0}. The Vignéras multiple gamma function can be regarded as a special case of the Barnes multiple gamma function. Namely G n z Γ n z,,, n the normalization factor..5 The q-gamma function Throughout this paper, we suppose 0 < q <. A q-analogue of the gamma function was introduced by Jackson [Jac04], [Jac05]. Γz ; q q z k q zk q k. This representation corresponds to the Weierstrass product representation of gamma function. Γz ; q has product representation like the Gauss product and like the Euler product..7.8 [] q [2] q [N] q Γz ; q lim [N ] z N [z ] q [z 2] q [z N] q. q Γz ; q { [n z } ]q [z n]q, [n] q [n] q n Askey [Ask78] pointed out that this function satisfies a q-analogue of the Bohr- Morellup theorem. Namely, Γz; q satisfies Γz ; q [z] q Γz; q, 2 Γ; q, 3 d 2 log Γz ; q 0 for z 0, dz2 where [z] q : q z / q. As q tends to 0, Γz; q converges Γz uniformly with respect to z. A rigorous proof of this fact was given by Koornwinder [Koo90]. Moak[Moa84], derived a representation of the q-gamma function log Γz; q z q z.9 2 log q 4

12 log q z q ξ ξ q ξ dξ C q 2 log q m R 2m z; q, k 2k B 2k log q M2k q z 2k! q z where C q 2 log q log q 2 q R 2m z; q 0 B 2m t 2m! 0 log q q tz 2 log q q t q t dt, B 2 t 2 2m M2mq tz dz, B n t : B n t {t} {t} denotes the Gauss symbol, i.e. the integral part of t., and the polynomial M n x is defined by the recurrence relation.0 M x, x 2 x d dx M n x nx M n x M n x. Each term of the formula.0 converges uniformly as q 0. So we get another proof of the uniformity of the classical limit of log Γz; q. 5

13 Chapter 2 A q-analogue of the multiple gamma function 2. Construction of a q-analogue of the multiple gamma functions In this section, we define a q-analogue of the multiple gamma functions which satisfy a q-analogue of the generalized Bohr-Morellup theorem, and we derive the expressions corresponding to.2,.3. We assume 0 < q <. Definition 2.. Let z be in the right half plane {s C Rs > 0} and r Z 0, we define G 0 z ; q : [z ] q, G n z; q : q z n q zk q k where k g n z, u : r kn 2 n z u u n n q k gnz,k, for n For example, G z; q is Γz; q. The infinite products of these functions are absolutely convergent, First, we prove a q-analogue of the generalized Bohr- Morellup theorem. Theorem 2..2 If Rz > 0, then G n z : q satisfy G n z ; q G n z; qg n z; q, 2 G n ; q, 3 d n dz n log G nz ; q 0 for z 0, 4 G 0 z; q [z] q. The functions satisfying such properties are determined uniquely. Proof: First we remark next formulas. 6

14 Lemma 2..3 g n 0, k 0. 2 g n z, k g n z.k g n z, k n kn 3 n 2 These formula can be proved by direct calculation. Next we prove the claim of theorem. When n, claim is the Bohr-Morellup theorem. So it is sufficient to show the case n 2. 2,4 can be proved easily. So we prove and 3. We can see G n z ; q q z n n z n { q zk n kn 2 n n kn 3 n q k n kn 2 n g nz,kg n z,k n kn 3 n 2 } q z n k q zk q z n k G n z; qg n z; q. 3 We can see d n q k q zk q k n kn 3 n 2 n kn 2 n dz n log G nz ; q { n dn } k n 2 q zkl dz n n l k l k n 2 log q n l n q zkl n 0, k l. q k gn z,k q k gnz,k because log q < 0. Finally, we prove the uniqueness of these functions. Because of the formula in the proof 3, d n dz n log G nz ; q 0 as z. Hence the uniqueness follows from Dufresnoy and Pisot s theorem [DP63]. By Theorem 2..2, G n z; q can be defined when z l m log q/2π l ; negative integer, m Z. 2.2 Infinite product representation of G n z ; q Next we consider expression like the formulas.2,.3. 7

15 Proposition 2.2. If Rz > 0, then { G n z ; q lim N { 2 G n z ; q k G n ; q G n N; q G n z ; q G n z N; q G n k; q G n z k; q n m Gn m k ; q G n m k; q n m } G n m N ; q m z. z m }. Proof: First we prove next formulas Lemma For n 2 For n 2, N, n m G n N ; q q N n N z N m n m 3 For N, k, n 2, n m N k n m N k k { z k n q k n N k. } k. n z N {g n z l, k g n l, k} g n z, k. m l They are shown by induction. We prove of Proposition By Lemma 2.2.2, { } G n ; q G n N; q n lim G n m N ; q m z N G n z ; q G n z N; q m { N lim q l {zl n n } l n N N m n m m z n z N N k l k q k n m n m N k m z N {gn zl,k gn l,k} l kn By Lemma , 3 and l0 k q k N l {gn zl,k gn l,k} }. q zkl n ln 3 n 2 q kl q zk k q k n kn 2 n, q zkl q kl n kn 3 n 2 8

16 we obtain { G n ; q G n N; q n lim N G n z ; q G n z N; q m n q zk q z q k k { lim q k n m n m m N k z N kn So, it is sufficient to show kn n kn 2 n q k n m m N k z as N. } G n N ; q m z }. q k gnz,k We can see { log q k n m m} N k z z N k m log q k n m kn kn z m k Nk N k N n m q k n m! q kn { z } q N m k n m k n m 2 k.kq k, n m! q k where we take the principal value of logarithms. This tend to 0 as N on any region, since n m! q k n m k n m 2 k.k.q k k takes finite value. Thus, { log kn Hence the claim follows. q k n m m} N k z 2 By using, we can prove the claim easily. 0 as N. 9

17 Chapter 3 The Classical Limit of the Multiple q-gamma Functions 3. The Euler-MacLaurin expansion of G n z ; q By means of the Euler-MacLaurin summation formula 3. N rm fr N M ftdt n k B k k! { } f k N f k M n N M B n t f n tdt for f C n [M, N], n! we give an expansion formula of the multiple q-gamma functions which we call the Euler-MacLaurin expansion. This formula plays an important role in the following sections. Proposition 3.. Suppose Rz > and m > n, then log G n z ; q { z n n n r n j0 r B r r! d r } z q z log dz n q { d r } z z ξ r q ξ log q dz n r! q ξ dξ m G n,j zc j q r B r r! F n,r z; q R n,m z; q, 0

18 where [ d r F n,r z; q : n C j q : n n! B r dt r r [ d r f j,r q : { t q zt log n q z r! f j,r q [ B n t dt r R n,m z; q : m m! { d n dt n { q t j t log q { t j log }] [ { d m B m t dt m The polynomial G n,j z is introduced through, t }] q t q { t n n z u G n,j zu j. n j0, t }}] dt, q zt log q z }}] dt. Proof: From 2.7 and the definition of G n,j z, we obtain 3.2 log G n z ; q z k log q log q zk n n n k G n,j z{ k j log q k }. j0 k Applying the Euler-MacLaurin summation formula, we have 3.3 k k log q zk n t log q zt dt n Let L r z and L r be m r m r B r r! { r } d t log q z dt n t B r r! F n,r z; q R n,m z; q. L r z : Li rq z log r q, L z : log q z, L r : L r q,

19 where Li r z is Euler s polylogarithm Li r z : k z k k r. From it follows that 3.4 log q zt l t log q tz dt n q tzl l, n j0 j n S j n! j r0 r j! j r! L r2z, where n S j is the Stirling number of the first kind defined by n ns j u j u n, j0 where u n uu u n. Substituting 3.4 to 3.3, we have 3.5 Similarly, k k log q zk n n j0 m r m r j n S j n! B r r! j r0 r j! j r! L r2z { r } d t L dt n t B r r! F n,r z; q R n,m z; q. 3.6 k j log q k k j r0 r j! j r! L r2 j r B r r! j! j r! L C j q. L r z and L r cause divergence as q 0, but we can prove that these divergent terms vanish. In order to simplify such terms, let us express L r z as the sum of L r and convergent terms. 2

20 Lemma 3..2 where L l z zl q z l! log q l z l r L r l r! r0 T r z z l r ξ r q ξ log q r! q ξ dξ. l z l r T r z, l r! Proof: By partial integration, we have 3.7 z ξ n qξ log q q ξ dξ n r0 n! r { z n r! log r n r Li r q z Li r q }. q Let L k z be Then 3.7 reads L k z Li kq z Li k q log k. q 3.8 n L n z k0 n k z n k L k z n k! n k0 n k z n k L k n T n z. n k! In order to write L n z with L n and T k z, we use induction on n. It can be seen that L n z zn L z n! n k0 z n k n k! L k which is equivalent to the claim of Lemma n k k z n k T k z, n k! Applying 3.5, 3.6 and Lemma 3..2 to 3.2, we obtain 3.9 log G n z ; q n j0 j n S j n! j l l j! j l! l r0 z l r l r! L r2 G n,j z n j0 j r0 r j! j r! L r2 3

21 { z m n n j0 n j0 r B r r! j n S j n! n r d t dt n t j l0 G n,j z j0 l n S j n! j l0 l j! j l! j r0 B r r! z l l! j! j r! L l j! z l j l! l n j B j d t j! dt n j0 t q z log q n j0 j n S j n! j l0 l j! j l! l r z l r T r z l r! r0 m r0 n B r r! F n.r z; q G n,j zc j q R n,m z; q. j We prove that the coefficients of the divergent terms in 3.9 vanish. By the following lemma, we can see that the first term and the second term in 3.9 are canceled out. Lemma 3..3 n j0 j n S j n! j l l j! j l! l r0 z l r l r! L r2 n G n,j z j0 j r0 r j! j r! L r2. Proof: We consider the exponential generating functions of the coefficients of L r2 in both sides. The generating function of the left hand side is n j0 j n S j n! j l0 l j! j l! l r0 z l r u r l r! r! n j0 j n S j z u j n! 4

22 z u. n On the other hand, the generating function of the right hand side is n j r j! u r G n,j z j r! r! j0 r0 n G n,j z u j j z u. n Therefore, the coefficients of L r2 in both sides coincide. Thus, the claim follows. and Next, we prove the coefficient of L vanishes. Since n j0 j n S j n! n G n,j z j0 n r0 n r0 n r0 j l0 j r0 B r r! B r r! B r r! l j! j l! B r r! z l l! j! j r! z n j! G n,j z j r! jr n G n,j z jr 0 t dt n { } r d t j dt t { r } d z t dt n t we have n r0 B r r! { d r } z, dz n the coefficient of L z z t m dt n n 0 5 r0 B r r! d r z dz n

23 m r0 B r r! d r z. dz n z0 We prove that the right hand side of the above formula is equal to zero. In a formal sense, e d dz d dz r B r r! d dz r. Imposing the boundary condition at z 0, we make the both sides above act on z n. Then, e d z dz n z 0 t dt n n r B r r! [ d r ] tz t 0 dt n t0 because t n is a polynomial of n -degree. Since F z : z n satisfies z F 0 0 F z F z, n it can be seen that e d z z dz. n n Thus we have the formula z 3.0 n z 0 t dt n n r B r r! [ d r ] tz t 0, dt n t0 which shows that the coefficient of L is equal to zero. Hence we have proved that the all coefficients of L r vanish in 3.9. Finally, we calculate the coefficients of log q z and T r z. Using the formula 3.0, we have 3. the coefficient of log q z q z n n r q in 3.2 B r r! d r z. dt n In order to calculate the coefficients of T r z in 3.9, we note that z u n n j0 n S j z uj n! n j0 j n S j n! j l0 l j! j l! l r z l r r0 l r! u r r!. 6

24 Using this identity, we have 3.2 the coefficient of T r z in 3.9 r d z u d r z. dt n dz n u0 Substituting 3. and 3.2 in 3.9, we obtain Proposition The classical limit of G n z ; q and the asymptotic expansion of G n z In this section, we study the classical limit of G n z ; q using the Euler- MacLaurin expansion. We will see that this limit formula gives an asymptotic expansion for the multiple gamma functions, which is a generalization of the Stirling formula for the gamma function The classical limit of G n z ; q First we consider the classical limit of the Euler-MacLaurin expansion in the domain {z C Rz > }. Proposition 3.2. Suppose Rz > and m > n. lim log G nz ; q q 0 { z n B r n r! n r n j0 r d r } z logz dz n { d r } z dz n r!r {z r } m G n,j zc j R n,m z, r B r r! F n,r z where n C j : r B r r! r d {t j log t} dt n n! t B n t n d { t j log t } dt dt 7

25 r { } d t z t F n,r : log, dt n z t R n,m z : m m! m { } d t z t B m t log dt. dt n z Furthermore this convergence is uniform on any compact set in {z C Rz > }. Proof: Taking Proposition 3.. into account, we must show that q z 3.3 lim log log z, q 0 q 3.4 z ξ r q ξ log q lim q 0 r! q ξ dξ ξ r r! dξ r!r {z r }, lim F n,r z; q F n,r z, q 0 lim R n,mz; q R n,m z, q 0 lim C jq C j, q 0 and further have to show that this convergence is uniform. Here we prove only 3.6. The other formulas can be verified in a similar way. Since lim q 0 d dt r { t log n } q zt q z r { d t log dt n } z t, z in order to show 3.6, it is sufficient to prove that the procedure of taking the classical limit commutes with the integration. Let us introduce polynomials M r x through d r dz r log qzt cf.[8]. They satisfy the recurrence relation. M x, r log q q zt q zt M r q zt x 2 x d dx M nx {r x }M r x M r x and M r r!. Using this we have m d t q zt B m t log dt n q z dt n! { n j j n S j j l0 m j l l m l log q B m tt j l q tz q tz M m lq dt} tz. 8

26 Therefore we have to show { m l log q 3.8 lim B m t t j l q 0 q tz q tz M m lq } tz dt { m l log q lim B mt t j l q 0 q tz q tz M m lq } tz dt. Lemma Suppose that α N be fixed and that y 0 and y are fixed constants such that < y 0 < y. Then there exists a constant C depending on y 0 and y such that α log q C 0 < q ty < t y α for y [y 0, y ], < t <, 0 < q <. Proof: Letting q e δ, we define { α tyδ ϕt, y, δ : e e δty δ > 0 δty δ 0. Then, ϕt, y, δ is positive, continuous and bounded on { } D : t, y, δ t, δ 0, y 0 y y. So we can put C : sup D ϕt, y, δ. Suppose < y 0 Rz y. By Lemma 3.2.2, there exists a constant C depending only on y 0 and y such that m l log q tj l q tz q tz M m lq tz C tj l t Rz m l for t <, 0 q <. Since m j 2, the right hand side above is integrable over t <. Therefore, Lebesgue s convergence theorem ensures 3.8. We have thus proved the limit formula 3.6. Next we show that this convergence is uniform on any compact set in the domain {z C Rz > }. Put m l log q Φz, q : B m tt j l q zt q tz M m lq zt dt. From the consideration above, we see that there exists a constant C depending only on y 0 and y such that Φz, q C B m t dt t 2 for Rz [y 0, y ] and 0 < q. 9

27 Hence {Φz, q 0 < q } is a uniformly bounded family of functions and Φz, q B m t tj l m l m l! t z m l dt as q 0. By Vitali s convergence theorem, this convergence is uniform on any compact set in the domain {z C Rz > }. The constant C j in Proposition 3.2. can be expressed in terms of the special value of the Riemann zeta function. Lemma C j ζ j j 2. Proof: From the definition of ζs, ζ s k log k k s for Rs >. By the Euler-MacLaurin summation formula, we obtain 3.9 n ζ s s 2 r B r r! r d { t s log t } dt t Since n! n d { B n t t s log t } dt. dt n d {t s log t} dt s s nt s n s n t s n log t, 3.9 can be analytically continued to {z C Rz > n }. So if we put s j, n j 2, then we obtain j2 ζ j j 2 r j j 2 2 B r r! j 2 C j. r l d { t j log t } dt t j2 d B j2 t {t j log t}dt dt Next we prove that the limit function in Proposition 3.2. coincides with the multiple gamma function. 20

28 Theorem Suppose m > n. Then, as q 0, G n z ; q converges to G n z uniformly on any compact set in the domain C\Z <0 and 3.20 log G n z { z n n r B r r! d r } z logz dz n n r n j0 R n,m z. { d r } z dz n r!r {z r } { G n,j z ζ j } j 2 m r B r r! F n,r z Proof: In the domain {z C Rz > }, we have already proved the existence of the limit function and uniformity of the convergence. Let us put G n z : lim G nz ; q. q 0 Because of the uniformity of the convergence, we have particularly [ n n d d { lim {log G n z ; q}] log q 0 dz dz G n z } so that, from Theorem 2..2, G n z satisfies the conditions in Theorem.4.. Namely Gn z G n z G n z 2 n d log dz G n z 0 for z 0 3 Gn 4 G0 z z. Since a hierarchy of such functions is uniquely determined, so G n z G n z in {z C Rz > }. Thus the claim of the theorem in the case when Rz > has been proved for {z C Rz > }. Next, we show that in {z C Rz, z }, G n z ; q G n z as q 0 and that the convergence is uniform on any compact set in this domain. For the proof, we use induction on n. The case when n was considered by Koornwinder [Koo90]. Let K be a compact set in {z C 2 < Rz, z }. If q is sufficiently close to then [z ] q 0 on K. Therefore as q 0, Γz ; q 2 Γz 2; q [z ] q

29 uniformly converges to on K. We assume that Γz 2 z Γz G n z ; q G n z as q 0 and that the convergence is uniform on K From the definition of multiple q-gamma function, we see that, if q is sufficiently close to, G n z ; q has no poles and no zeros on K, neither has G n z. Therefore, as q 0, uniformly converges to G n z ; q G nz 2; q G n z ; q G n z 2 G n z G nz on K. Repeating this procedure, we can verify, for any n, G n z ; q converge to G n z in a compact set in the domain { 3 < Rz 2, z 2}, { 4 < Rz 3, z 3},. Thus the claim of the theorem is proved Asymptotic expansion of G n z Let us call the expression 3.9 the Euler-MacLaurin expansion of G n z. We should note that 3.9 is valid for z C\{Rz }. We show that it gives an asymptotic expansion of G n z as z, i.e. the higher Stirling formula. Theorem Let 0 < δ < π, then log G n z { z n n n r n j0 r B r r! d r } z logz dz n { d r } z dz n r!r {z r } { G n,j z ζ j } j 2 r B 2r 2r! F n,2r z as z in the sector {z C argz < π δ}. 22

30 Proof: Straightforward calculation shows that F n,r z r r l l r l n { d dt r l } t n t n S k k k r l k0 { } l d logz t dt t l l! z l. Thus, if r > n, then F n,r z Oz rn as z. So we can see that 3.2 F n,2r z F n,2r 3 z Oz 2 as z. Furthermore we can see that R n,2m z Noting that 2m! n 2m! j j n S j 2m { } d t z t B 2m t log dt dt n z j j l m l! l0 z t < t sin δ t j B 2m t t z 2m j dt. in the sector {z C argz < π δ} and that B 2m t B 2m for 0 t, we have 2m { } d t z t B 2m t log dt dt n z Hence, B n 2m 2m! j as z in the sector. j n jl 2m l! n S j l sin δ 2m j l0 R n,2m z Oz 2m n of n,2m z dt t 2m j. Let us exhibit some examples of the higher Stirling formula. when n, we obtain In the case log G z log Γz z logz z ζ 0 2 r B 2r 2r 2 z 2r. 23

31 This is the Stirling formula since ζ 0 2 log2π. Furthermore, in the case when n 2, we obtain log G 2 z z 2 2 logz z2 z 2 4 zζ 0 ζ 2 z B 2r z 2r, 2r 3 z 2r r2 which coincides with the formula.5. In the case when n 3, 4, and 5, we have the following results. Proposition The higher Stirling formula for n 3, 4 and 5 are as follows: log G 3 z z 3 z2 z 2 6 z logz 36 z z2 z ζ 0 2z ζ 2 2 ζ 2 2 z B 2r { z 2 2r 4 z 2r 6r z 4r 2 6r 6 }. log G 4 z z 4 r2 24 z3 6 z logz 4 72 z4 2 9 z3 z z 3 44 z3 3z 2 2z ζ 0 3z2 6z 2 ζ z ζ 2 6 ζ 3 2 z 720 z 3 6z z 5 2 r3 B 2r { z 3 2r 5 z 2r 2r 27z 2 20r 2 94r z 8r 3 56r 2 34r 09 }. 24

32 log G 5 z z 5 20 z z3 z logz z z z3 z z z4 6z 3 z 2 6z 24 ζ 0 4z3 8z 2 22z 6 ζ 24 6z2 8z ζ 2 2z ζ 3 24 ζ 4 2 z z 3 4 z z 9 2 r3 B 2r { z 4 2r 6 z 2r 20r 54 z 3 70r 2 375r 506 z r3 540r r 354 z 6r r3 754r r The Weierstrass product representation for the G n z By calculating the formula 3.20 in the case of m n, we derive the Weierstrass product representation for the multiple gamma functions. Main theorem of this section is the following: Theorem 3.3. For n N, we have where G n z exp F n z k n n 2 F n z : G n,j zq j z j0 z 0 Φ n z, k : z u du γ, n n! n 2 µ }. { z k } n exp Φn z, k, k r0 { n rµ [ r! u n S r r µ zr µ r ] uz z u ζ r n u0 } µ k µ, 25

33 Q j z : P j z P j x : ϕ j,r : j j r0 j r0 j j r r B r r! ϕ j.rx j r, j z r P j r r B j r z r l z l, l l r { } d t j dt j log t tj j 2. t Proof of this theorem will be carried out through Section , and some examples of this representation will be discussed in Section Rewriting the Euler-MacLaurin expansion of G n z In this section we prove the following proposition. Proposition n log G n z G n,j zk j z j0 where K j z : B jz logz ζ j P j z j [ P j z k P j z k B ] jz k z k log. j z k k Furthermore the infinite sum of the last term is absolutely convergent. Proof: From the definition of G n,r z, we have 3.23 d dt and hence r z d n du r z u r!g n,r z, n u n r { d r } z dz n r!r {z r } z0 n G n,j z z j j 2. j0 26

34 Since t n we have, for r n 3.25 n z z t G n,j zz t j, n j0 n F n,r z G n,j zz j r ϕ j,r. j0 In order to calculate the coefficient of logz, we use the following lemma. Lemma n z G n,j z B n j n j G n,j z B jz. j j0 Proof: Since the both sides are polynomials in z, it is sufficient to prove that the formula holds for a sufficiently large, arbitrary integer N. Noting that { B j N B N j l lj j > 0 j N j 0, we have n G n,j N B jn j j0 j0 n j0 N n G n,j Nl j G n,0 N l j0 N l N n N n N l N n n N. n G n,j N B j j From 3.23 and Lemma 3.3.3, it follows that z n B r d r z 3.26 n r! dt n r n z G n,j z B j n j j0 j0 n G n,j z B jz. j 27

35 Next we calculate R n,n z. From the definition of G n,r z, we have 3.27 R n,n z n n! d B n t dt n { z z t log n } z t dt z 3.28 n j0 j0 [ n G n,j z n! B n t n { } ] d z t z t j log dt dt z [ n n G n,j z n! k Here we have, for j n, 0 0 B n t n { d z t k j log dt z t k n { d B n t z t k j log dt Ok j n as k, z } ] dt. z t k z } dt so that the infinite sum in 3.27 converges absolutely because j n 2. By means of the Euler-MacLaurin summation formula, we have 3.29 n n! { 0 z k j { z k j j n { d B n t z t k j log dt z kj j j r j r z t k z B r r! j r z k j r B r r! j r z k j r [ z k j z k j z kj j j j n r0 r } } } dt logz k logz k B r r! j { r z k j r z k j r}] logz B r r! ϕ { j,r z k j r z k j r}. 28

36 The coefficient of logz of the above formula vanishes because of the following lemma. Lemma B j z k j z k j z k j j z kj j j r j r B r r! j r z k j r B r r! j r z k j r. Proof: The first equality follows from the identity j j B j z k z k j r B r. r r0 The second equality holds because of the Euler-MacLaurin summation formula for z t k j on [0, ]. Applying Lemma to 3.29, we obtain n [ Bj z k z k R n,n z G n.j z log j z k n r0 j0 k B r r! ϕ { j,r z k j r z k j r}]. If r j 2, then z k j r z k j r decreases more rapidly than k 2 as k. Thus we have n n B r 3.30 R n,n z G n,j z r! z j r ϕ j,r j0 rj2 { Bj z k z k log j z k k }] P j z k P j z k, and the infinite sum is absolutely convergent. Substituting 3.24, 3.25, 3.26, 3.30 to 3.20, and noting that n j 2 we obtain 3.2. r B r r! z j r n rj2 B r r! z j r P j z, The next lemma which will be useful in Section is deduced from the above proof. 29

37 Lemma B j z k j z k log z k Ok 2 as k. P j z k P j z k In other words, the polynomial P j z k P j z k is a convergent factor such that the infinite sum in 3.30 converges absolutely An infinite product representation for the ζ j We derive an infinite product representation for ζ j in the same way as in Section Proposition expζ j expp j k k and the infinite product converges absolutely. B j k j Proof: From the proof of Lemma 3.2.3, we have j2 ζ j j 2 j j 2! r P j B j2 j 2! ϕ j,j2 j j 2! B r r! ϕ j,r exp P j k P j k j2 d B j2 t {t j log t}dt dt j2 d B j2 t {t j log t}dt. dt Furthermore we can see by the same way as 3.28 that j2 d 3.3 B j2 t {t j log t}dt Ok 2 as k, dt Hence we have j j 2! j j 2! j2 d B j2 t {t j log t}dt dt k 0 j2 d B j2 t {t k j logt k}dt, dt and the infinite sum of this formula is absolutely convergent. In the same fashion as in the previous section, we have 3.32 j j 2! 0 B j2 t j2 d {t k j logt k}dt dt 30

38 B jk log P j k P j k j k B j2 j 2! k ϕ j,j2. k From 3.3 and 3.32, it follows that 3.33 Hence we have ζ j P j B j k j log P j k P j k k Ok 2 as k. log k k B j k j and the infinite sum is absolutely convergent. exp {P j k P j k}, Remark In a similar fashion to Proposition 5.6, an infinite product representation of ζ j, z : d ds ζs, z s j can be given as follows: expζ j, z exp zj j log z P jz B j zk z k j exp P j z k P j z k. z k k Good representation for K j z In this section, we give a good representation for K j z. The following two lemmas are useful in subsequent arguments. Lemma Let k be a positive integer and k, then we have k r log z k r l l z l k r l Ok 2 l 2 B j z k j Ok 2. log z j r B j r k j r0 l l z k l r l Proof: is clear. 2 follows from the identity j B j z k B j r z k r. r0 3

39 Lemma There exists a unique polynomial Ak, z such that B j z k j Ak, z is equal to P j z k P j z k. Proof: By integrating the both side of from to 0, we get the formula log z Ak, z Ok 2. k j B j z B j r z r r j j j r r0 B j r r r 0. From Lemma and 3.34, a polynomial Ak, z satisfying 3.35 B j z k j log z Ak, z Ok 2 as k, k is uniquely determined and the polynomial P j z k P j z k satisfies 3.35 by Lemma Therefore, Ak, z P j z k P j z k. Using these lemmas, we prove the following proposition. Proposition Let k be a positive integer and define and Then we have 3.36 I k : j { j } j r l z l B j r z k l r, r l r0 II k : I k, III k : IV k : r0 l j z j r {P r k P r k} z j k j, j r0 B j z k j { Bj z k j j r z j r r l z k log z k l z l k r l. l } k log I k z k P j z k P j z k 32

40 { } Bj z k z k log II k j k { } Bj z k k log III k j k { z k j log z } IV k k { j j r0 } r z r B j r z r k k Furthermore each term in the right hand side decreases like Ok 2 as k. Proof: Making use of the identity, B j z k B j z k j z k j, we have 3.37 B j z k j B jz k j B jz k j z k log z k k log B jz k z k log z k j k k log k z k j log z k k. By Lemma 3.3.8, 3.33 and the identity B j z k j j j j r r0 B j r zk r, we can see that each term in the right hand side of 3.36 is Ok 2. On the other hand, it can be seen that I k IV k are polynomials of z and that I k j { j } j B j r z r z r r r k r0 a polynomial of k, II k j { j } j B j r z r z r r r k r0 a polynomial of k, 33

41 Hence, if we put III k zj a polynomial of k, j k IV k { j zj r0 r j } j z j polynomial of k r k a polynomial of k. j k Bk, z : I k II k III k IV k { j } j r z r B j r z j r r r0 then, Bk, z is a polynomial of k, z and it satisfies B j z k j By Lemma 3.26, we have k k log z Bk, z Ok 2 as k. k, 3.38 Bk.z P j z k P j z k. By 3.37 and 3.38, we can deduce Proposition 3.3. K j z Q j z k j r j z j r ζ r zj r j γ { z k k log z k j r0 j r z j r r l and the infinite sum of this formula converges absolutely. } l z l k r l l Proof: The summation of each term in 3.36 from k to k, is absolutely convergent because of Proposition 5.9, and the following calculation is possible. Namely, { } Bj z k k 3.39 log I k j z k k { } Bj z k z k log II k j k k B jz j logz j 34 j j r l z l B j r z. r l r0 l

42 Noting Theorem and we have γ k { log }, k k 3.40 { Bj z k k j j r0 log } III k k j z j r {ζ r P r } zj r j γ. By 3.39 and 3.40, we obtain 3.4 { } Bj z k z k log P j z k P j z k j z k k B jz j j j r0 k j j r r logz B j r z r l z l l j z j r {ζ r P r } zj r j γ { z k j log z k j r0 l j r z j r r l } l z l k r l. l The proof is completed by substituting 3.4 to the definition of K j z in Proposition A proof of main theorem Using the results in Section 3.3. Section 3.3.4, we prove Theorem By Proposition 3.3., we have 3.42 log G n z [ n j j G n,j z Q j z z j r ζ r zj r j γ j k r0 { z k j log z k 35 j r0 j r z j r r l0 }] l z l k r l. l

43 It is easily seen that 3.43 n j j G n,j z z j r ζ r, r j r0 n 2 n G n,j zz j r ζ r, r0 n 2 r0 j0 [ r! u ] uz z u ζ r, n u n G n,j z zj j j0 z n G n,j zz k j j0 0 z u du, n k. n Substituting 3.43, 3.44, 3.45 to 3.42, we have 3.46 log G n z n n 2 G n,j zq j z j0 z 0 r0 z u du γ n G n,j z n j0 j r0 k [ r! u j r z j r r l r ] uz z u ζ r n u0 [ k log z n k l z l l k r l. Thus, in order to prove Theorem 3.3., it is sufficient to show 3.47 n G n,j z j0 n! j r0 r z j r n 2 µ l { n rµ l z l l k r l n S r r µ zr µ } µ k µ. Since 3.45 and z k j log z k j r0 Ok 2 as k, j r z j r r l l z l l k r l 36

44 we have 3.48 k log z n G n,j z n k j0 Ok 2 as k, while we obtain k log z 3.49 n k n! n 2 µ k log z n k { n rµ j r0 j r z j r r l n S r r µ zr µ } l z l l µ k µ n r r n S r k r l z l n! l k r0 Ok 2 as k, l k r l by Lemma We can deduce that 3.48 and 3.49 imply 3.47 by the same arguments as in Lemma Hence the proof is completed Examples of the Weierstrass product representation for G n z We give some examples of the Weierstrass product representation for the multiple gamma functions. In the case when n, we have G z Γz e γz k { z } e z k. k This is the Weierstrass product representation for the gamma function. In the case when n 2, we have G 2 z Gz e zζ 0 z2 2 γ z2 z 2 k { z } k exp z z2. k 2k Since ζ 0 2 log2π, this is the Weierstrass product representation for the Barnes G-function [Bar99] In the case when n 3, 4 and 5, we obtain the following results. 37

45 Proposition The Weierstrass product representations in the case when n 3, 4 and 5 are as follows: G 3 z exp { z3 4 z z ζ k } zz z ζ z2 4 γ [ z kk { 2 z 3 exp k 6 z2 z 2 4 k 4 z z } ] 2 2 k, G 4 z { 6 exp 44 z4 3 8 z z z z 2 ζ 2 z2 2z 3 k ζ z3 3z 2 2z 6 [ z { kkk2 6 z 4 exp k 24 z3 6 z2 6 k ζ 0 z4 4z 3 4z 2 } γ 24 z 3 8 z2 4 z z z k z6 }] 2 k2, G 5 z { exp z z z3 z z z 6 ζ 3 z2 3z 4 z4 6z 3 z 2 6z 24 k ζ 2 2z3 9z 2 z ζ 2 ζ 0 6z5 45z 4 0z 3 90z 2 γ 720 [ z kkk2k3 { 24 z 5 exp k 20 z z3 z2 8 k z 4 96 z z2 z z z2 8 k 24 z 2 24 z k 2 z24 }] 4 k3. } 38

46 Part II A q-analogue of the Hurwitz Zeta-Function 39

47 Chapter 4 A q-analogue of the Hurwitz Zeta-Function 4. Observation from the theory of the quantum group SU q 2 The left regular representation of the quantum group SU q 2 is, by definition, the coordinate ring ASU q 2 viewed as a left U q sl2-module. It is known that it has the following homogeneous decomposition. ASU q 2 where each homogeneous component corresponds to the irreducible representation of U q sl2 with spin n 2, and dima n n 2. The Casimir operator of U q sl2 C q k 2 qk 2 2 q q 2 ef n A n acts on each homogeneous piece as a scalar operator: n q 2 q n C An λ n q q. Thus, according to the standard argument of differential geometry, one can introduce a spectral zeta-function associated to the quantum group SU q 2 as follows: n Zs; SU q 2 λ s, n which reads, substituting 4. to 4.2, 4.3 n Zs; SU q 2 q 2s n n 2 q ns [n] 2s q These observations yields two interesting issues: The first one is to consider the spectral geometry of quantum groups, or in general, quantum homogeneous 40

48 spaces [MMN 90], [Pod87]. The second is to construct a q-analogue of zetafunctions of various types, and to study their analytic or arithmetic aspects. We think 4.3 tells us how to define a q-analogue of zeta-functions. For instance, let us consider the multiple zeta-functions [Bar00]: ζs, z; ω k,k 2, k n0 k ω k 2 ω 2 k n ω n z s. From 4.3, it seems quite natural to define a q-analogue by 4.4 ζs, z; ω, a, b; q k,k 2, k n0 q sak anknb [k ω k 2 ω 2 k n ω n z] q s. In this part, we restrict ourselves on the simplest case among these q-multiple zeta-functions. Namely we will consider only a q-analogue of the Hurwitz zetafunction cf.[kur92], q sk 4.5 ζs, z; q [k z] s q k0 and investigate its analytic property. The results in this paper except the functional relation can be generalized to the case of the following q-multiple zeta-functions ζ n s, z; q k,k 2, k n0 kn n k0 q skk2 kn [k k n z] s q q sk [k z] q s. As well as the q-hurwitz zeta-function, these q-zeta-functions are deeply linked to generalized hypergeometric functions and polylogarithm functions. 4.2 The q-hurwitz zeta-function The Hurwitz zeta-function 4.8 ζs, z k0 k z s is a meromorphic function in s. The pole of it is only s and simple. A q-analogue of this zeta-function is introduced by 4.9 ζs, z; q k0 q sk [k z] q s. We call it the q-hurwitz zeta-function. The right-hand side of 4.9 is absolutely convergent in the right half plane {s Rs > 0}, so it is holomorphic in this domain. It is easy to show that, if Rs >, the classical limit q 0 of the q-hurwitz zeta-function coincides with the Hurwitz zeta-function. 4

49 Let us consider analytic continuation of this q-zeta-function. By making use of the binomial expansion theorem, we can rewrite the right-hand side of 4.9 to ζs, z; q q q 2 s s r q rz 4.0 r! q rs, where s r ss s r. Since 0 < q <, Rz > 0, the right hand side above is absolutely convergent for r0 4. s r δl, where r runs over the set of non-negative integers Z 0, l runs over the set of integers Z and δ 2πi log q.. This implies that ζs, z; q has meromorphic continuation to the whole s-plane. The poles are s r δl r Z 0, l Z, and simple. It is not difficult to calculate the first few coefficients of the Laurent expansion at s 0. Thus we obtain Proposition 4.2. The q-hurwitz zeta-function ζs, z; q has meromorphic continuation to the whole s-plane. The poles are s r δl r Z 0, l Z and all simple. The Laurent expansion at s 0 reads as follows: { } 4.2 where ζs, z; q α s α 0 s α log q z k k α log q, α 0 2 logq q2 log q α 2 log q 2 logq q2 log 2 q q 2 2 log q and we have put log k x log x k. Os 2, We should notice the following fact: As mentioned before, the pole of the Hurwitz zeta-function is only s, which is simple. In particular, it is holomorphic at s 0. On the other hand, the poles of the q-hurwitz zeta-function are s r δl as pointed out in the proposition. Namely, in the classical limit, the poles of the q-hurwitz zeta-function drastically changes. 4.3 The Euler-MacLaurin expansion We apply the Euler-MacLaurin summation formula 3. to the study of the q-hurwitz zeta-function by putting ft is expressed as qts q tz s f k t q st sp kq tz ; s q tz sk logk q The k-th derivative of ft where P k x; s are polynomial of x defined by the following recursive relation P 0 x; s s, P x; s, 42

50 x x 2 P kx; s kx sp k x; s P k x; s. It is found that P k x; s is a polynomial of degree k in x and also degree k in s with positive integral coefficients, and that sp k ; s s k, sp k 0; s s k. Putting these ingredients into the Euler-MacLaurin summation formula, we obtain the following expansion formula. Proposition 4.3. When Rs > 0, we have ζs, z; q q q2 s s log q F s, s, s ; qz q q 2 s q z q q 2 s m 2k B 2k log q s q z 2k! q z P 2k q z ; s where 4.4 R 2m s.z; q s 0 R 2m s, z; q B 2m t 2m! k and F α, β, γ; x is a hypergeometric function. 2m s log q q q tz q tz q st P 2mq tz ; sdt We call 4.4 the Euler-MacLaurin expansion of ζs, z; q. The appearance of a hypergeometric function in 4.4 is due to the integral formula Γγ 4.5 F α, β, γ; x ΓβΓγ β 0 u β u γ β xu α du. Now we consider the way that the q-hurwitz zeta-function function is expressed as an infinite series of hypergeometric functions. Putting m in 4.4, we get ζs, z; q q q2 s s log q F s, s, s ; qz q q 2 s 2 q z s q q 2 s log q 2 q z q z R 2 s, z; q, where the residue term R 2 s, z; q is defined by 4.4. Note that 4.6 B 2 t π 2 n cos2πnt n 2. Substituting this into 4.4 and making use of the integral formula 4.5, we obtain the following expansion formula. Theorem ζs, z; q q q2 s s log q F s, s, s ; qz q q 2 s q z sq q2 s 2πi ls δl F s, s δl, s δl; qz. l 0 The infinite series in the right-hand side is absolutely convergent for s r δl r Z 0, l Z, l 0 so that it is holomorphic in this region. 43

51 4.4 Classical limit Using Theorem4.3.2, we can achieve profound understanding for the classical limit of ζs, z; q. As was mentioned before, for Rs >, lim ζs, z; q ζs, z. q 0 In this section, we will consider in detail the classical limit of ζs, z; q in the complement of this region. A key to this consideration is the Gauss connection formula of a hypergeometric function, 4.8 F α, β, γ; x ΓγΓα β γ x γ α β F γ α, γ β, γ α β ; x ΓαΓβ ΓγΓγ α β F α, β, α β γ ; x. Γγ αγγ β Applying this to F s, s, s ; q z and F s, s δl, s δl; q z in 4.7, we get lim {ζs, z; q q q2 s q zs } π 4.9 q 0 log q sin πs z s s z s 2 sz s 2πi l 0 U, s; 2πilz l where l ranges over the set of non-zero integers, and Uα, γ; x is a solution of the confluent hypergeometric differential equation satisfying, for Rα > 0, 4.20 x d2 y dy γ x dx2 dx αy 0 Uα, γ; x x α as x. It is written, in terms of confluent hypergeometric functions F α, γ; x, as 4.2 Uα, γ; x Γ γ F α, γ; x Γ α γ Γγ e x x γ F α, 2 γ; x, Γα and has the following integral representation [Sla47]: 4.22 Uα, γ; x Γα 0 e xu u γ α du. Note that the right-hand side of 4.20 is absolutely convergent unless s is a positive integer because of the asymptotic behavior

52 Applying the Euler-MacLaurin summation formula to the Hurwitz zetafunction, we have 4.23 ζs, z z s s z s 2 s ss 2 z s 2! 0 B 2 t dt. z t s2 Using 4.6 and 4.22, we rewrite the integral term above. Then we see the right-hand side of 4.20 coincides with Namely, 4.24 Thus we obtain ζs, z z s s z s 2 sz s U, s; 2πilz. 2πi l l 0 Theorem 4.4. If s is not a positive integer, lim {ζs, z; q q q2 s q zs } π 4.25 ζs, z. q 0 log q sin πs 4.5 A q-analogue of the functional equation for the Riemann zeta-function Let us put ζ s, z; q ζs, z; q q q2 s q zs 4.26 π log q sin πs. This is rewritten, as mentioned in the previous section, as follows: ζ q q 2 s q z 4.27 s, z; q s q z F,, 2 s; q z log q q q 2 s 2 q z Ss, z; q T s.z; q where 4.28 and 4.29 Ss, z; q q q 2 s zs Γ s q 2πi l l 0 T s, z; q 2πi q q 2 s q z l 0 Γs δl e 2πilz Γδl l F δl,, s; qz. The right-hand side of 4.28 is absolutely convergent for s r δl r are positive integers and l are non-zero integers and that of 4.29 is so for Rs < 0. We calculate T s, z; q under the assumption that 0 < z, Rs <, by means of the integral representation 4.22 and 4.30 we get T s, z; q s q q 2 B t π q z Thus the following theorem is deduced: n sin2πnt n s q z log q F,, 2 s; qz 2 q q 2 s q z. 45